Question

if A B C D are four collinear points then eulers thearm states

Answer 1

Answer:

In geometry, the Euler line, named after Leonhard Euler (/ˈɔɪlər/), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.[1]

Euler's line (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).

The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron.

Step-by-step explanation:

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Answer 2

Collinear points out Euler's theorem

• According to Euler’s theory, the one-dimensional points and exceeding triangle exist as a mathematician line in a very triangle, within which 3 points of concurrency of Triangulum lie. The 3 points on the Euler’s line are the orthocenter, the circumcenter, and also the center of mass.

• Collinear points: These are points that line on a similar line; any 2 points are continually one-dimensional points by drawing a straight line. 3 or additional points are one-dimensional however don’t have to be compelled to air a similar line.

First method:

For the space formula, we'd like to calculate the space between purpose A and purpose B, then the space between purpose B and purpose C, and check if the add (AB + BC) of those 2 distances is up to the space between purpose A and purpose C.

By the distance formula, if the first principle could be a line then, AB + BC=AC

Second method :

If you're given three points, A(x,y,z) ,B(a,b,c),C(p,q,r)

Find the space between AB = =√(x-a)^2 + (y-b)^2 + (z-c)^2,

AB + BC=AC, then points are one-dimensional

Third method:

Use the idea that the world of Triangulum fashioned by 3 one-dimensional is zero. a technique is by determinants.

Fourth method:

The direction ratios of 3 vectors a,b, and c are proportional; they're one-dimensional.

Example:

Show that the 3 points P(4, 6), Q(6, 8), and R(8, 10) are one-dimensional.

Solution: The definition says, if the given 3 points P(4, 6), Q(6, 8), and R(8, 10) are one-dimensional, then the slopes of any 2 pairs of points PQ, QR & PR are equal.

Now, we will realize the slopes of several pairs of points by employing a slope formula such that:

Slope of PQ = (8 – 6)/ (6– 4) = 2/2 = one

Slope of QR = (10 – 8)/ (8 – 6) = 2/2 = one

Slope of PR = (10 – 6) /(8 – 4) = 4/4 = one

From the top of the example, the slopes of all the pairs of points are equal.

Hence it says, the 3 points P, alphabetic character, and R are one-dimensional.

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